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  "body": "\n## Summary\n\nA measure-preserving transformation is a measurable map between [[measure spaces|P15VQS5DJ9]] under which the preimage of every measurable set has the same measure as the set.\nEquivalently, the [[pushforward|3N66TM3TSQ]] of the first measure is the second one.\nA measure-preserving map of a space to itself is the basic object of ergodic theory.\nIt models a system that evolves in time without creating or destroying mass, as in Hamiltonian mechanics.\n\n## Explanations\n\n### L1 human\n\nThink of a fluid in a container that is stirred, where no fluid is added or lost.\nAfter one step of stirring, the fluid that ends up in a region came from a region of the same volume.\nA rotation of a circle preserves length, so it is measure-preserving.\nThe doubling map $x \\mapsto 2x \\bmod 1$ on $[0,1)$ is also measure-preserving, although it stretches: each interval has two preimage intervals of half the length.\n",
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